P. Marcolongo
∗ON THE SOLVABILITY IN GEVREY CLASSES OF A LINEAR
OPERATOR IN TWO VARIABLES
Abstract. We show non solvability results in Gevrey spacesGsfor a linear partial differential operator with a single real characteristic of constant multiplicity m, m≥3, provided s>m/(m−2)+δ, whereδ >0 depends on the order of the degeneracy of a suitable lower order term. In particular, δ→0 as the order of the degeneracy tends to+∞.
1. Introduction
The main aim of the present paper is to study in detail the local solvability of a model linear partial differential operator in two variables. We recall that although most of the well known classical operators, appearing in the basic theory of PDEs and in Mathe-matical Phisics, are solvable, non solvable operators exist, as proved first by Lewy [9], and often of a very simple form. The example of Lewy was generalized by H¨ormander [6], who proved a necessary condition for the local solvability of partial differential operators, given by the following
THEOREM 1. Let the linear partial differential operator P with coefficients in
C∞()be solvable in, in the sense that for every f ∈ C∞0 ()we can find a so-lution u∈D′()of Pu= f . Then, for every compact set K ⊂there exist a positive constant C and an integer M≥0 such that
(1)
Z
f(x)ϕ(x)d x≤ X
|α|≤M
sup
x∈K
|Dα f(x)| X
|α|≤M
sup
x∈K
|Dα tPϕ(x)|
for all f ,ϕ∈C0∞(K).
If an operator P is non locally solvable in the previous sense it is natural to analyze its behaviour in Gevrey spaces, being intermediate classes between the analytic and the
C∞functions. More precisely, from now on we will refer to the following functional setting.
Let K be a compact subset ofand C a positive fixed constant. Let us consider the subspaceG0s(,K,C), 1<s<+∞, of C0∞(), given by all the functions f(x)with ∗The author is thankful to Professor Luigi Rodino and Professor Petar Popivanov for useful discussions
and hints on the subject of this paper.
support contained in K such that, for some R≥0,
(2) sup
x∈K
|∂αf(x)| ≤R C|α|(α!)s.
Gs
0(,K,C)is a Banach space endowed with the norm: (3) kfks,K,C:=sup
α
C−|α|(α!)−s sup
x∈K
|∂αf(x)|
or equivalently
(4) kfks,K,C:=
X
α
C−|α|(α!)−sk∂αfkLp(K)
with p≥1 fixed. From now on we consider (4) with p=2. DEFINITION 1. Gs
0() =
S
K,CG0s(,K,C)where K and C run respectively
over the set of all the compact sets contained inand over the set of the positive real numbers.
Therefore it is natural to endowGs
0()with the inductive limit topology:
(5) ind lim
Kր
Cր+∞
Gs
0(,K,C).
Similarly we defineGs()as the projective limit topology of the spacesGs(,K,C) of all the functions f ∈C∞()for which the norm (3) of the restriction to K is finite. The main result that we will handle in the following, concerning with the solvability in a Gevrey frame of partial differential operators withGs coefficients, is the Gevrey version of Theorem 1 proved by Corli [2].
THEOREM 2. Let s be a fixed real number, 1<s <+∞. Let the linear partial differential operator P be s-solvable in, i.e. for all f ∈ G0s()there exists u in
D′
s(), space of the s-ultradistributions in, solution of Pu = f . Then for every compact subset K of, for everyη > ǫ > 0, there exists a positive constant C such
that:
(6)
max
x∈K|f(x)|
2
≤Ckfk
s,K,η−1ǫ k
tP fk s,K,η−1ǫ
for every f ∈Gs 0(,K,
1 η).
The previous theorem will be used in the following manner: an operator P is not
Now let us analyze the problem we are interested in. Let us consider the operator
(7) P =Dmx
1 −A(x1)D
m−1
x2 −B(x1)D
m−2
x2 Dx1
with m odd, m ≥3; A(x1), B(x1)are analytic functions of x1, defined in a neighbor-hood of x1=0.
The operator P is weakly hyperbolic in x1with a single characteristic of constant mul-tiplicity m and therefore it is always locallyGs solvable for s < mm−1 without any restrictions on A(x1)and B(x1). This well known fact follows from theGswell posed-ness of the Cauchy problem for P or from more general results about theGs solvability of linear PDEs with multiple characteristics (e.g., cf. Corollary 5.1.3 in Mascarello-Rodino [13], see also [1], [5], [17]). Next if ℑm(A(x1)) vanishes of odd order at
x1=0, changing its sign from - to +, the results of Corli [3] imply that P is not locally solvable inGs for s > m
m−1. We will investigate the case ofℑm A(x1)vanishing of even order at x1=0.
Let us suppose that for some integer h>0:
ℜe A(0)6=0, (8)
ℑm A(x1)=cx12h+o(x12h) for x1→0, c6=0. (9)
Moreover for a fixed l>0:
(10) ℑm B(x1)=d x1l +o(x1l) for x1→0, d6=0.
Popivanov [14] ( see also Popivanov-Popov [15]) proved that P is not locally solvable, in the C∞sense, if h is sufficiently large with respect to l.
Moreover, if the conditions (8), (9), hold, as a particular case of Theorem 3.7 in [10] or Theorem 3.2 in [4] we obtain that the operator (7) is s-solvable for
(11) s< m
m−2.
At this point a natural question arises: what can we say about the behaviour of P for indexes s≥ mm−2? We get that P is non s-solvable for
(12) s> m
m−2+θ (h), withθ (h)→0 for h→0.
The proof of this result, developed in the next section, is based on Theorem 2, applied to a function fλ(x)of the following form:
fλ(x)=v(x)·ei λψ0(x)+λ
m−1
m ψ1(x)+λ
m−2
m ψ2(x)+...+λ
1
mψm−1(x)
.
We will choose the phase functionsψi(x)according to a standard proceeding (see [3]).
functionv(x)such that the approximate solution f(x)of the homogeneous equation
tP f
λ =0 makes the right hand side of (6) arbitrarily small forλsufficiently large and
s satisfying (12), but leaves the left hand side greater than or equal to 1. We observe
that the previous estimates need also a suitable version for this case of the results of Ivrii [8]. For more details we also address to [12].
2. The main result
THEOREM3. Let us suppose that A (respectively, B) satisfies (8), (9) (respectively, (10)) and
(13) h≥
2(2l+2) if m=3 2l+2 if m≥5.
Then P is non s-solvable at the origin for every s satisfying (12) where
θ (h)= m(l+1)
(m−2)[(2m−4)h−(m−1)l−1].
Proof. We will reason ab absurdo.
First let us observe that
tP = −(−i)m∂m x1−(−i)
m−1A(x
1)∂xm2−1−(−i)
m−1B(x
1)∂xm2−2∂x1+
−(−i)m−1B′(x1)∂xm2−2.
(14)
Let us define
(15) fλ(x):=ei λψ0(x)+λ
m−1
m ψ1(x)+λ
m−2
m ψ2(x)+...+λ
1
mψm−1(x)
where x=(x1,x2).
1st STEP: choice of the phase functionsψi(x).
The basic idea is to apply the operatortP to fλ(x)and choose the phase functions ψ0(x),ψ1(x),ψ2(x)in order to make equal to zero the higher powers ofλ.
Following a standard approach for constructing formal asymptotic solutions (e.g., cf. [2], see also [5], Chapter IV) we chooseψ0=x2and obtain
tP f
λ(x)=
lm−1(x)λm−1+o(λm−1)
fλ(x), λ→ +∞.
where the coefficient is given by
(16) lm−1:= −
∂ψ1(x) ∂x1
m
−A(x1).
Letting lm−1=0, we obtain m complex solutions and we choose
(17) ψ1(x)=
Z x
0
e
where eA(x1) = [ A(x1)]
Repeating the previous arguments with this choice ofψ1(x)finally we find:
(19) ψ2=
witheB(x1)satisfying again (10) for a new constant d. Therefore
(20) ℑmψ2(x)=
The other phase functions are determined recursively, according to a standard proceed-ing, cf. [2]. We need not precise information on them, but they vanish at x1=0. We then write According to our choices we find:
tPu(x)= f
is an analytic function;
- Qj(x,D)are differential operators of order less than or equal to m, with analytic
3rd STEP: minimum ofℑm8(x).
We follow here the arguments in Popivanov [14]. Obviously
(24) ℑm8(0,0)=0.
We are interested in the solutions of the following equations: ∂ℑm8(x)
The second equation is solved by
(27) x2=o(λ−
1
mx1), for x1→0.
Let us try to solve the first equation λ Taking largeλ, in this way we have to solve
(28) cx12h+ 1
From the previous definition it follows immediately thatλ− 1
m =ǫ2h−l; thus, replacing this quantity in (28) we obtain the following equation:
(30) cx12h+ 1
then (30) transforms into
Now let us consider the corresponding function
Then, by the Implicit Function Theorem there exists a function y1=y1(ǫ)∈C2, such that g(y1, ǫ)=0, y1(0)=0, y1′(0)=0. This implies that
Therefore, considering (29) and (27) we conclude that a critical point is given by
x1λ=
Now, we may give a complete picture of the behaviour ofℑm8(x)varying l and sign of d:
- d·c > 0: if d,c > 0ℑm8(x)is an increasing function of x1; otherwise
ℑm8(x)is a decreasing function of x1.
We may limit our attention to the case l odd and d <0 (the case l even, c>0, d <0 is treated in the same way. Note moreover that the change of variable x2′ = −x2gives a change of sign for the constant d).
We want to apply Taylor’s formula in order to analyze the behaviour ofℑm8(x)near the point(x1λ,x2λ). To this aim let us observe that: Now let us suppose that
we get
C0′(x1−x1λ)2λδ2+ 2hX+1
j=3
(x1−x1λ)jλ
m−1
m −
2h+1−j m(2h−l) =
=C0′(x1−x1λ)2λδ2{1+O(ǫ1)}. (39)
Now let us define:
(40) g1(x1):=
(
1 if x1∈[1−ǫ21,1+ǫ21] 0 if x1∈[1−ǫ1,1+ǫ1]. Let us suppose that:
- g1(x1)∈G0s(R);
- 0≤g1(x1)≤1, for every x1inR. By definition, it follows immediately that
g1(x1x−1λ1)=1 if |x1−x1λ| ≤ ǫ1
2x1λ and
su pp g1(x1x1λ−1)⊂ {x1∈R : |x1−x1λ| ≤ǫ1x1λ}. Moreover
su pp∂∂x
1g1(x1x
−1
1λ)⊂ {x1 : (1−ǫ1)x1λ≤ |x1−x1λ| ≤(1− ǫ1
2)x1λ}
[
{x1 : (1+ǫ21)x1λ≤ |x1−x1λ| ≤(1+ǫ1)x1λ}. Then on su ppg1(x1x1−λ1), and for|x2−x2λ| ≤ǫ1we have
(41) ℑm8(x)≥Ce(λ)+C0′λδ2(x
1−x1,λ)2[1+O(ǫ1)]+ 1
2|x1−x1λ| 2λmm−2.
In an analogous way on su pp∂g1
∂x1 and for|x2−x2λ| ≤ǫ1we get
(42) ℑm8(x)≥Ce(λ)+C0′′λδ1
with
(43) C0′′:=ǫ12C0′
− d mc
2
2h−l >0 and 0< δ1<1, defined as before.
Moreover let us consider:
(44) g2(x2):=
(
- g2(x2)∈G0s(R);
- 0≤g2(x1)≤1, for every x1inR. Let us suppose that x2∈su pp g2. Then
|x2−x2λ| ≤ ǫ21 + |x2λ|.
Considering that x2λ→0 forλ→ +∞, forλsufficiently large we have
|x2−x2λ| ≤ǫ1.
Moreover, by definition we have
su pp∂g2(x2)
∂x2 ⊂ {
ǫ1
4 ≤ |x2| ≤ ǫ1
2}.
Then, on su pp∂g2(x2)
∂x2 , for|x1−x1λ| ≤ǫ1x1λand forλsufficiently large we obtain
(45) ℑm8(x)≥Ce(λ)+ǫ12λ m−2
m .
4t hSTEP: choice of v(x). Let us define
(46) χ (x1,x2):=g1(x1x1−λ1)g2(x2).
Now let us observe that the Cauchy Kovalevsky Theorem assures us the existence of functionsvλ(k)defined by induction as solutions of the following transport equations:
(47)
Q0(x,D)v(0)λ =0
Q0(x,D)v(λk) = −P
j λ−
j
mQj(x,D)v(k−j)
λ , k≥1 vλ(0)=1 on x1=x1λ
vλ(k)=0 on x1=x1λ for k≥1 where in the sum j runs from 1 to min{k,m−1}. Let us consider:
(48) Vλ(N) :=
N
X
k=0 v(λk).
Let us define
(49) uλ(x):=χ (x)Vλ(N)(x)e
i8(x)
5t hSTEP: the conclusion.
We want to apply the necessary condition of Corli [3] to uλ(x). We recall that we want to contradict the estimate (6) for f =uλ,λ→ +∞, i.e.
(50) max
K |uλ(x)|
2
≤Ckuλk
s,K,η−1ǫk
tPu
λk
where K is a suitable small compact neighborhood of the origin. In view of (46), we may limit our attention to
(51) K := {(x1,x2) : |x1−x1λ| ≤ǫ1x1λ, |x2| ≤ ǫ21}. Let us start evaluating the left hand side.
Let us observe that
(52) |uλ(x1λ,x2λ)| =e−ℑm8(x1λ,x2λ)=e−eC(λ), therefore
(53) max
K |uλ(x)|
2
≥e−2Ce(λ).
Now let us analyzekuλk
s,K,η−1ǫk
tPu
λk
s,K,η−1ǫ.
Let us reason as in Ivrii [8]: there exist positive constants B and N such that, setting λ=4BeLN, L≥1,
the following estimate holds:
(54) X
|α|≤m
kDαvλ(k)k
s,K,1η ≤Be
−Lk+4M N ν
m
, k=0,1,2, . . .
withν= mm−1. Therefore
X
|α|≤m
kDαVλ(N)k
s,K,1η ≤2Be 4M N
ν
m .
Since our choice ofvλ(k)implies
tP
λVλ(N) =
mX−2
h=0
mX−2
j=h λ−
h
mQhvN−m+2+j
λ ,
from (54) it follows that
(55) ktPλVλ(N)k
s,K,1η ≤Ce
−L N+4M N ν
m .
Now
kuλk
s,K,η−1ǫ ≤ kχks,K,η+1ǫ kV (N) λ ks,K,1
η
kei8k s,K,η+1ǫ
≤c e4M N
ν
m kei8k
s,K, 1 η+ǫ
.
PROPOSITION1. Let9be an analytic function in a neighborhoodof the origin
Applying the previous proposition we obtain that:
kei8k sition 1, considering also (42), we obtain that:
kei8k
therefore:
1≤C1e−L N+8M N ν
m+dλ 1
s(η 1
s+(η+ǫ) 1
s)
+C2λ
m−1
m e8M N
ν
m−C′′0λδ1+dλ 1
s(η 1
s+(η+ǫ) 1
s), ∀N.
Clearly the first addend of the right hand side tends to zero when N→ +∞.
Regarding the second addend of the right hand side, in order to let it to zero when
N → +∞, we must impose
ν m < δ1
(60)
1
s < δ1.
(61)
Recalling that
δ1=
m−1
m −
2h+1
m(2h−l)
we get that (60) is satisfied if the hypothesis (13) is valid and (61) is equivalent to require s>mm−2+θ (h). This concludes the proof.
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AMS Subject Classification: 35A07.
Paola MARCOLONGO Dipartimento di Matematica Universit`a di Torino via Carlo Alberto, 10 10123 Torino, ITALIA
e-mail:[email protected]
